Illusions can produce sounds.
If you wish to use spells other than thaumaturgy, you can try these illusion spells:
- Minor Illusion
You create a sound or an image of an object within range that lasts for the duration. The illusion also ends if you dismiss it as an action or cast this spell again.
If you create a sound, its volume can range from a whisper to a scream. It can be your voice, someone else’s voice, a lion’s roar, a beating of drums, or any other sound you choose. The sound continues unabated throughout the duration, or you can make discrete sounds at different times before the spell ends.
The description specify that you can create a sound that is as loud as a scream. While not technically superhuman, the loudest scream recorded according to Guiness comes to 129 decibels which is comparable to a jet engine. A message sustained at that volume will carry for quite a distance.
- Programmed Illusion
You create an illusion of an object, a creature, or some other visible phenomenon within range that activates when a specific condition occurs. The illusion is imperceptible until then. It must be no larger than a 30-foot cube, and you decide when you cast the spell how the illusion behaves and what sounds it makes. This scripted performance can last up to 5 minutes.
When the condition you specify occurs, the illusion springs into existence and performs in the manner you described. Once the illusion finishes performing, it disappears and remains dormant for 10 minutes. After this time, the illusion can be activated again....
The volume created under Programmed Illusion isn't specified but an imaginative commander might use this spell to issue regular announcements loudly.
- Major Image
You create the image of an object, a creature, or some other visible phenomenon that is no larger than a 20-foot cube. The image appears at a spot that you can see within range and lasts for the duration. It seems completely real, including sounds, smells, and temperature appropriate to the thing depicted. You can’t create sufficient heat or cold to cause damage, a sound loud enough to deal thunder damage or deafen a creature, or a smell that might sicken a creature (like a troglodyte’s stench). As long as you are within range of the illusion, you can use your action to cause the image to move to any other spot within range. As the image changes location, you can alter its appearance so that its movements appear natural for the image. For example, if you create an image of a creature and move it, you can alter the image so that it appears to be walking. Similarly, you can cause the illusion to make different sounds at different times, even making it carry on a conversation, for example...
Volume for this is limited to under the threshold of thunder damage. This can be interpreted as extremely loud. If this article is accurate, infra-sound in the range of >151 decibels would cause "humans... to feel their chest moving without their control", taking that as the threshold of thunder damage, you can still produce a sound that is louder than actual thunder. Your general's voice could easily boom over a city at this volume.
Customising spells and/or magic items
If these do not suit your needs the DMG has guidelines on creating magic items (pg 284) or spells (pg 283)
As BBeast has pointed out:
The DMG has instructions on customising magic items by adding special features like minor properties (DMG p 143) to an item. The Minor Property 'War Leader' achieves your desired effect.
War Leader. The bearer can use an action to cause his or her voice to carry clearly for up to 300 feet until the end of the bearer's next turn.
The sidebar "Creating Additional Common Items" on page 140 of Xanathar's Guide to Everything suggests you can use these effects to create new common magic items.